Most traditional board and card games, such as Chess, Chinese Chess, Go, Chinese Mahjong, Hearts, Bridge, etc., share the same playing model: Players play around tables using physical objects such as cards and may hold objects in their own private areas, e.g., players hold cards in their own hands in Bridge. In this paper, this model is called the play-on-table (POT) game model and these games following the model are called POT games. The research of this paper is summarized as follows. First, formalize the definition of the POT game model. Second, present some game systems to allow players to design and play new POT games. Third, prove that these game systems are general for all POT games. Finally, in order to demonstrate the theory, practically implement one of the general game systems that allows players to design and play new POT games in a what-you-see-is-what-you-get (WYSIWYG) manner.